In this note, some recurrence relations are derived between the expected values of functions of order statistics from any arbitrary distribution with continuous cumulative distribution function (cdf). These recurrence relations are closely related to some results obtained by Srikantan .
"A Note on Recurrence Relations between Expected Values of Functions of Order Statistics." Ann. Math. Statist. 37 (3) 733 - 734, June, 1966. https://doi.org/10.1214/aoms/1177699471